

Title of Thesis
Generalization of Bayesian Approach of Testing
Many Hypotheses as Conditional Optimization Problem 
Author(s)
Abdul Mueed 
Institute/University/Department
Details Abdus Salam School of Mathematical Sciences / GC
University, Lahore 
Session 2011 
Subject Mathematics 
Number of Pages 112 
Keywords (Extracted from title, table of contents and
abstract of thesis) Methods, Algorithms, Bayesian,
Problem, Generalization, Optimization, Calculation, Results,
Hypothesis, Conditional, Approach, Testing, Computation, Special,
Case 
Abstract Multiple hypothesis
testing is an important topic in statistics.Therefore, the problem
addressed in this thesis is an important one.It is also a topic in
which it is difficult to make a significant improvement, for various
reasons.One reason is that often different users may have different
objectives and with multiple hypotheses there is no unique objective
function. In the thesis is recognized this fact and as the objective
functions, estimated the quality of made decisions, are used
minimization of the probabilities of the errors of one kind at
restrictions of the probabilities of the errors of second kind. Such
approach is a new one which causes the uniqueness of the regions of
acceptance of hypotheses and, consequently, improves the quality of
hypothesis testing.
Thus conditional Bayesian tasks of testing many hypotheses are
stated and solved.The concept of conditionality is used for
designation of the fact that the Bayesian tasks are stated as
conditional optimization problems where the probabilities of onetype
errors are restricted and, under such conditions, the probabilities
of secondtype errors are minimized.The properties of obtained
decision rules are investigated, and, on their basis, it is shown
that the classical Bayesian problem of hypotheses testing is a
special case of the considered.The calculation results of concrete
examples have shown that the qualities of offered conditional tasks
surpass the quality of the classical Bayesian task.They completely
confirm the results of theoretical investigations. The convenience,
simplicity and naturalness of introduction of similar gradation
Kiefer,(1977) by the level of certainty of hypotheses testing on the
basis of concrete observation result are shown in offered
conditional tasks.
Quasioptimal procedures of many hypotheses testing are offered.They
significantly simplify Bayesian algorithms of hypotheses testing and
computation of the risk function.The obtained general solutions are
reduced to concrete formulae for multivariate normal distribution of
probabilities.The methods of approximate computation of the risk
functions in Bayesian tasks of testing many hypotheses are offered.
The properties and interrelations of the developed methods and
algorithms are investigated. On the basis of simulation, the
validity of the obtained results and conclusions made is shown.
The results of sensitivity analysis of the conditional Bayesian
problems are given and their advantages and drawbacks are
considered. 
